feat(NumberTheory/Transcendental/Liouville): irrational reals are LiouvilleWith for 1 < p ≤ 2

[localecho]

Closes the gap noted in this file's module doc — the reverse implication for 1 < p ≤ 2 was flagged 'not formalized yet' since 2021. Proved via Dirichlet approximation (Real.infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational) plus a finite-fiber argument on denominators, rather than continued fractions directly.

---

Formalized with Claude's help, verified by a clean lake build.

Posted about this gap in #new-members first: https://leanprover.zulipchat.com/#narrow/channel/113489-new-members/topic/Intro.3A.20Brigham.20Hall.20.3B.20LiouvilleWith.20gap.20for.201.20.3C.20p.20.E2.89.A4.202/near/622672679

[github-actions]

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[github-actions]

PR summary a337177272

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith	2292	2294	+2 (+0.09%)

Import changes for all files

Files	Import difference
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith Mathlib.NumberTheory.Transcendental.Liouville.Measure	2

---

Declarations diff (regex)

+ Irrational.liouvilleWith
+ finite_den_eq_and_close
+ infinite_den_of_irrational

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit a337177).

• +1 new declarations
• −0 removed declarations

+Irrational.liouvilleWith

---

No changes to strong technical debt.
No changes to weak technical debt.

Current commit a337177272
Reference commit b7f093a35e

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.py pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[metakunt]

Suggested change

	· have := hx.ne_rational q.num (n : ℤ)
	simpa using this
	· simpa using hx.ne_rational q.num (n : ℤ)

[SnirBroshi]

by and exact are inverses

Suggested change

	have hn' : (0 : ℝ) < n := by exact_mod_cast hn
	have hn' : (0 : ℝ) < n := mod_cast hn

(this doesn't always work, but hopefully will)